If I have durable power of attorney for my mother, do I have the right to stay in her house after she passes away?

In my state (DE) POA becomes void after death. Check your mom's will to see if she had any inclusions about her possessions and home

Once a person dies power of attorney dies with them. The will governs with the executor making decisions.

You can always make your brother a reasonable offer for his share of the house, when the time comes. But the will stands.

In her will, who is named as executor?
If its you, then that is ideal situation for you to continue to live at the house & do whatever is needed to transfer property as per her will which is a 50/50 split. If its both you & bro to be co-executors, it's going to be more involved and imo you'll need an atty.

What I'd suggest for you to do now is to review her will to see exactly what she wants done. Then do some research into how property can be distributed after death for your state. States vary as to how this can be done in probate court. Like if your state allows for a "muniment of title"; a "small estates affadavit"; or if you need to open full traditional probate. Some definitely need an atty while others maybe not. But this way you'll have a better idea of what will be needed and can speak more clearly with your brother as to the approach to take.

Personally in my not an atty viewpoint, I think if house & car are owned outright, the assessor value of house is sensibly accurate & mom died with no real debts; & you have all her documents; you are named executor & have your wits about filing & finding things at the courthouse; & bro releases any heirship; then you can totally do a Muniment of Title on your own to get will done & house in your name. Muniment costs vary but its likely to quite a bit less than full probate & likely under $1k and maybe with 3 filings at courthouse within a set # of months (like 6-8 mos). If there's all sorts of debts (claims), or if mom was on Medicaid, or you get flummoxed going to the courthouse or placing a notice in the paper or your brother starts challenging your staying in & getting the house, then you will need to hire an probate atty whether it's muniment, small estate or full probate needed.

Also about determining assets of her estate, the value of the house, its contents, value of car, all other funds and any insurance policy named beneficiary will add up to be the value (assets) of her estate. This $ amount is important as small estates affadavit will require it be under a specific amount. You don't want it to happen that mom died and was $10k above the small estates limit and mom could have put a new roof on the house for 11k.

Thanks everyone for your input on this matter - great advice on a subject unknown to me. In Mom's will, my brother are listed as co-personal representatives (executors?) as well as co-beneficiaries of assets and property. To complicate things, Mom is on Medicaid in the nursing home - her house is the only asset not dealt with already. All her monetary assets went into a personal maintenance account for my Mom/house maintenance but in my name as POA.
Will look into this with my Medicaid planning attorney and get her take on things.